706 MECHANICS Geome trical theor em. Cycloi- tlal mo tion. Hence the speeds of M and M are as MO : OM , and therefore, by what we have stated above about elementary arcs at M and M , the proximate position of MM is also a tangent to B, which proves the proposition. It is easily seen from this that, if one polygon of a given number of sides can be inscribed in one circle and circum scribed about another, an infinite number can be drawn. For this we have only to suppose a number of particles moving in A with speeds due to a fall from L, and then if they form at any time -the angular points of a polygon whose sides touch B they will continue to do so throughout the motion. Fig. 44 shows two forms of a quadrilateral possessing this property. Tauto- chrone. Resisted motion of pen dulum. Fig. 44, 137. To find the time of fall from rest doivn any arc of an inverted cycloid. Let O (fig. 45) be the point from which the particle commences its motion. Draw OA parallel to CA, and on BA describe a semicircle. Let P,P , P" be corresponding points of the curve, the generating circle, and the circle just drawn, and let us compare the speeds of the particle at P and the point P", Let P"T be the tangent at P". Speed of I " Speed of P = element at P" element at P ^p//T^p//T /A_ B__ATB_ / ~~ BP ~ BP"/ AB " 2AT 7 V . A B AH But speed of P = /(1g . A M). /_?. V A B Hence speed of P"= /errs A B, a constant. And, as the length of A F B is JTT . A B, time from A to B in circle = time from to B in cycloid AB /AB V 2< The time of fall to the vertex from all points of the curve is therefore the same. Hence the cycloid is called a " tautochrone." 138. As an instance of cases in which the acceleration depends upon the speed and the position jointly, take the motion of a simple pendulum in a medium whose resistance varies as the velocity directly. This is "the law, at least approximately, for very small speeds, whether the pendulum oscillate in a gas or in a liquid, and even when the resistance is due to magneto-electric induction, as when the pendulum is a magnetic needle vibrating in presence of a conducting plate or a closed coil. A syn thetical solution of this problem has already been given under Kinematics in 68, Analytically : if I be the length of the string, its deflexion from the vertical at time t, m the mass of the bob, we have evidently mW= - _ The ratio it/ml may be increased (theoretically) without limit by increasing the surface which the bob exposes, without changing its mass. But it cannot be indefinitely diminished. We will write 2k for it. If we assume the angle of oscillation to bo small, we may write the equation in the form where n^^g/l, and k is essentially positive, being greater as the resistance (whether on account of the viscosity of the medium or the large surface of the bob in proportion to its mass) is greater. A particular integral of this equation is evidently e=- p( i provided or p = - , Hence there are two quite distinct cases of motion, distinguished by differeut/orms of solution, depending on the relative magnitudes of k and n. These are separated from one another by the unique case in which k=n. (a) Let k > n, and let & 2 - n 2 = n 2 < k 2 . Then both values of p are real and negative. Pit ,-, Vn> = A + Eg " Thus If A and B have the same sign, diminishes, without changing sign, as t increases. But if A and B have different signs, the factor in brackets may vanisli for one definite value of t, and then change sign. After that the whole reaches a maximum and then dimin ishes without further change of sign. Examples of these eases are furnished (1) when the pendulum is displaced from the vertical and allowed to fall back ; it then approximates asymptotically to its position of equilibrium ; and (2) when it is drawn aside and flung back ; in this case it may pass once through the position of equilibrium and then asymptotically return to it. (Z>) Let n > k and let ?t 2 - k" = ?i 2 < ?t 2 . Here both values of p are imaginary, and we have This may be looked upon as a "simple harmonic motion" ( 52), of which the amplitude diminishes in a geometric ratio with the time, the decrement depending on the resistance alone, while the Seriod is permanently lengthened in the ratio n: n . This ratio epends both upon the original period and the resistance, so that for the same medium and same bob it is different for strings of different lengths. This investigation gives an approximation to the gradual dying away (by internal friction or by imperfect elasticity, &c. ) of all vibratory movements. The rate of diminution of amplitude, say of torsional vibrations of a wire, is thus a valuable indication and measure of a somewhat recondite physical quantity, which, without this method, would (at present at least) be hard to measure. (c) AVhen n = k, i.e., in the transition case, the equation becomes whose interal is known to be This, also, ultimately diminishes indefinitely as t increases ; but, as in case (a), it may either do so continuously or after having once passed through the value zero and reached a maximum, according to the relative magnitude and the signs of A and B. 139. When the path is given, the determination of the motion under given forces is, as we have seen, a mere question of integration of the equation for acceleration along the tangent. But more is required if we wish to find the normal pressure on the constraining curve. This is at once supplied by compounding the resolved part of the applied forces in the direction perpendicular to the tangent, with the additional force mV 2 /p acting from the centre of curvature. But, strictly speaking, all such questions require the application of Law III. Kinetics of a Particle ivith Tivo Degrees of Freedom. 140. The simplest case is that of a projectile, when rroj gravity is supposed to be uniform and to act in parallel tilc lines throughout the whole path, and the resistance of the le air is neglected. This has been sufficiently discussed in 40-42. It is merely the combination of (1 ) the uniformly accelerated motion of a stone let fall, with (2) a uniform velocity in a definite direction. Looked at from this point of view, it gives an interesting example of the graphic method applied in 53 to indicate the nature of simple harmonic motion. 141. We can extend these projectile results so as to
take account of the alteration of direction and of intensity